Simplify - i xa-xb a2-ab-b2- xb-xc b2-bc-c2- xc-xa c2-ca-a2 ii xa-x-b a2-ab-b2- xb-x-c b2-bc-c2- xc-x-a c2-ca-a2

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Byju's Answer

Standard IX

Mathematics

Quotient Law

Simplify : i ...

Question

Simplify :

(i) ${(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}}$

(ii) ${(\frac{x^{a}}{x^{- b}})}^{a^{2} - a b + b^{2}} \times {(\frac{x^{b}}{x^{- c}})}^{b^{2} - b c + c^{2}} \times {(\frac{x^{c}}{x^{- a}})}^{c^{2} - c a + a^{2}}$

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Q. Prove that:

(i)

{(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}} = 1

(ii)

{(\frac{x^{a}}{x^{- b}})}^{a^{2} - a b + b^{2}} \times {(\frac{x^{b}}{x^{- c}})}^{b^{2} - b c + c^{2}} \times {(\frac{x^{c}}{x^{- a}})}^{c^{2} - c a + a^{2}} = 1

(iii)

{(\frac{x^{a}}{x^{b}})}^{c} \times {(\frac{x^{b}}{x^{c}})}^{a} \times {(\frac{x^{c}}{x^{a}})}^{b} = 1

Prove that :

(i) ${(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + a b + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}} = 1$

(ii) ${(\frac{x^{a}}{x^{b}})}^{c} \times {(\frac{x^{b}}{x^{c}})}^{a} \times {(\frac{x^{c}}{x^{a}})}^{b} = 1$

Q. Simplify :

{(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}}

Q. Simplify :

{(\frac{x^{a}}{x^{- b}})}^{a^{2} - a b + b^{2}} \times {(\frac{x^{b}}{x^{- c}})}^{b^{2} - b c + c^{2}} \times {(\frac{x^{c}}{x^{- a}})}^{c^{2} - c a + a^{2}}

Q. The value of

{(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \cdot {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \cdot {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + b^{2}}

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Simplify : (i) (xaxb)a2+ab+b2×(xbxc)b2+bc+c2×(xcxa)c2+ca+a2 (ii) (xax−b)a2−ab+b2×(xbx−c)b2−bc+c2×(xcx−a)c2−ca+a2